Theorems REF , REF , and REF  are closely related to the Elekes-Rónyai theorem, which we will now briefly describe. In [1]}, Erdős and Szemerédi proved that if \(A\subset \mathbb {R}\)  is a finite set, then either the sum set \(A+A=\lbrace a+a^\prime \colon a,a^\prime \in A\rbrace \)  or the product set \(A.A=\lbrace aa^\prime \colon a,a^\prime \in A\rbrace \)  must have cardinality much larger than that of \(A\) .
